#517275

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(9,10,11,12)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,28,27,23);(1,2,4,3)(6,8)(14,19)(15,20)(21,25)(22,27);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,28)(26,27);(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,25)(26,28);(1,3,4,2)(13,17,18,26)(14,21,19,25)(15,22,20,27)(16,23,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,e^2=gi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424471);